Simplify the expression.
$\frac{b-9}{b^2-81}$
Select the correct choice below and fill in any answer boxes in your choice.
A. $\frac{b-9}{b^2-81}=$ $\square$ (Simplify your answer.)
B. The expression cannot be simplified.
Answer (2)
Factor the denominator as a difference of squares: b 2 − 81 = ( b − 9 ) ( b + 9 ) .
Rewrite the expression: b 2 − 81 b − 9 = ( b − 9 ) ( b + 9 ) b − 9 .
Cancel the common factor ( b − 9 ) : ( b − 9 ) ( b + 9 ) b − 9 = b + 9 1 .
The simplified expression is: b + 9 1 .
Explanation
Understanding the Problem We are given the expression b 2 − 81 b − 9 to simplify.
Factoring the Denominator First, we need to factor the denominator. Notice that the denominator b 2 − 81 is a difference of squares. We can factor it as follows: b 2 − 81 = ( b − 9 ) ( b + 9 ) So, we can rewrite the original expression as: b 2 − 81 b − 9 = ( b − 9 ) ( b + 9 ) b − 9
Simplifying the Expression Now, we can cancel the common factor ( b − 9 ) from the numerator and the denominator, assuming b = 9 :
( b − 9 ) ( b + 9 ) b − 9 = b + 9 1 Thus, the simplified expression is b + 9 1 .
Final Answer Therefore, the simplified expression is b + 9 1 .
Examples
Simplifying rational expressions is a fundamental skill in algebra, with applications in various fields. For instance, in physics, you might encounter such expressions when dealing with electrical circuits or fluid dynamics. Imagine you're designing a circuit where the resistance is given by a rational expression. Simplifying this expression can help you optimize the circuit's performance and reduce unnecessary complexity. Similarly, in economics, simplifying rational expressions can aid in analyzing cost functions or revenue models, leading to better decision-making in business scenarios.
The simplified expression of b 2 − 81 b − 9 is b + 9 1 . This is obtained by factoring the denominator and canceling the common factor. Thus, the answer is b + 9 1 .
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